Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pgrple2abl | Structured version Visualization version GIF version |
Description: Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.) |
Ref | Expression |
---|---|
pgrple2abl.g | ⊢ 𝐺 = (SymGrp‘𝐴) |
Ref | Expression |
---|---|
pgrple2abl | ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → 𝐺 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pgrple2abl.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | 1 | symggrp 17643 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
3 | 2 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → 𝐺 ∈ Grp) |
4 | 2nn0 11186 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
5 | hashbnd 12985 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 2 ∈ ℕ0 ∧ (#‘𝐴) ≤ 2) → 𝐴 ∈ Fin) | |
6 | 4, 5 | mp3an2 1404 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → 𝐴 ∈ Fin) |
7 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
8 | 1, 7 | symghash 17628 | . . . 4 ⊢ (𝐴 ∈ Fin → (#‘(Base‘𝐺)) = (!‘(#‘𝐴))) |
9 | 6, 8 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → (#‘(Base‘𝐺)) = (!‘(#‘𝐴))) |
10 | hashcl 13009 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0) | |
11 | 6, 10 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → (#‘𝐴) ∈ ℕ0) |
12 | faccl 12932 | . . . . . 6 ⊢ ((#‘𝐴) ∈ ℕ0 → (!‘(#‘𝐴)) ∈ ℕ) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → (!‘(#‘𝐴)) ∈ ℕ) |
14 | 13 | nnred 10912 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → (!‘(#‘𝐴)) ∈ ℝ) |
15 | 11, 11 | nn0expcld 12893 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → ((#‘𝐴)↑(#‘𝐴)) ∈ ℕ0) |
16 | 15 | nn0red 11229 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → ((#‘𝐴)↑(#‘𝐴)) ∈ ℝ) |
17 | 6re 10978 | . . . . 5 ⊢ 6 ∈ ℝ | |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → 6 ∈ ℝ) |
19 | facubnd 12949 | . . . . 5 ⊢ ((#‘𝐴) ∈ ℕ0 → (!‘(#‘𝐴)) ≤ ((#‘𝐴)↑(#‘𝐴))) | |
20 | 11, 19 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → (!‘(#‘𝐴)) ≤ ((#‘𝐴)↑(#‘𝐴))) |
21 | exple2lt6 41939 | . . . . 5 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐴) ≤ 2) → ((#‘𝐴)↑(#‘𝐴)) < 6) | |
22 | 11, 21 | sylancom 698 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → ((#‘𝐴)↑(#‘𝐴)) < 6) |
23 | 14, 16, 18, 20, 22 | lelttrd 10074 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → (!‘(#‘𝐴)) < 6) |
24 | 9, 23 | eqbrtrd 4605 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → (#‘(Base‘𝐺)) < 6) |
25 | 7 | lt6abl 18119 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (#‘(Base‘𝐺)) < 6) → 𝐺 ∈ Abel) |
26 | 3, 24, 25 | syl2anc 691 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ≤ 2) → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℝcr 9814 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 6c6 10951 ℕ0cn0 11169 ↑cexp 12722 !cfa 12922 #chash 12979 Basecbs 15695 Grpcgrp 17245 SymGrpcsymg 17620 Abelcabl 18017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-tset 15787 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-eqg 17416 df-symg 17621 df-od 17771 df-gex 17772 df-cmn 18018 df-abl 18019 df-cyg 18103 |
This theorem is referenced by: (None) |
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